THE VLA-COSMOS PERSPECTIVE ON THE INFRARED–RADIO RELATION. I. NEW CONSTRAINTS ON SELECTION BIASES AND THE NON-EVOLUTION OF THE INFRARED/RADIO PROPERTIES OF STAR-FORMING AND ACTIVE GALACTIC NUCLEUS GALAXIES AT INTERMEDIATE AND HIGH REDSHIFT

The VLA-COSMOS Project (Schinnerer et al. 2004, 2007) has imaged the COSMOS field at 1.4 GHz with the VLA to a mean sensitivity of ≲0.01 (∼0.04) mJy beam⁻¹ at the center (edge) of the field. As the basis of our subsequent analysis, we use the VLA-COSMOS "Joint" Catalog (Schinnerer et al. 2010) which contains ∼2900 sources detected with signal-to-noise ratio (S/N)⩾ 5. Nearly 50% of these are resolved at a resolution (FWHM of synthesized beam) of 25 × 25. Flux measurements were carried out with Astronomical Image Processing System (AIPS; Greisen 2003) and have been corrected for bandwidth smearing in the case of the unresolved radio sources. Their errors are generally ∼17% of the flux value.

IR data at 24 and 70 μm were taken by the S-COSMOS Survey (Sanders et al. 2007) using MIPS on Spitzer.

At 24 μm the FWHM of the point-spread function (PSF) is 58 and the average 1 σ sensitivity is ∼0.018 mJy over a large fraction of the imaged area. Flux measurements (LeFloc'h et al. 2009) in the deep and crowded MIPS 24 μm image were performed with the PSF-fitting algorithm DAOPHOT (Stetson 1987) which can simultaneously fit and hence de-blend multiple sources. In the following, we use a flux-limited catalog of COSMOS 24 μm sources which is restricted to the range S_ν(24 μm)⩾ 0.06 mJy. Typical flux uncertainties at 24 μm are ∼8%.

MIPS 70 μm observations of the COSMOS field were carried out in parallel with the 24 μm imaging. They have an average 1 σ point source noise of 1.7 mJy and a resolution of 186. Our 70 μm source list includes detections down to S/N = 3 and was compiled as described in Frayer et al. (2009). Fluxes were derived using the Astronomical Point source EXtraction (APEX; Makovoz & Marleau 2005) peak fitting algorithm; their average uncertainty is ∼16%.

In Figure 1, we show the minimum total IR (TIR; 8–1000 μm) luminosity that is detectable as a function of redshift given the sensitivity of the 1.4 GHz (converted to an IR measurement assuming the local IR–radio relation) and 24/70 μm data. Different IR spectral energy distribution (SED) template libraries (see color scheme in the lower right corner) lead to somewhat different predictions but it is clear that the 1.4 GHz and 70 μm surveys have matching depths, while the 24 μm observations are about seven times deeper. A similar sampling of the IR luminosity function is achieved in all three bands if a 24 μm flux limit of approx. 0.3 mJy is assumed. We therefore limit our 24 μm catalog to the range S_ν(24 μm)⩾ 0.3 mJy when we construct our IR-selected sample but allow fainter counterparts of 1.4 GHz sources to be included in the radio-selected sample. The use of these flux-limited samples has the immediate consequence that we only detect the brightest ULIRGs at z≳ 1.5, while the average luminosity of our sources is much lower at, e.g., z∼ 0.5, where most sources belong to the LIRG class.

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Optical data and photometric redshifts are taken from the COSMOS photometry catalog of Ilbert et al. (2009) which lists more than 600,000 COSMOS galaxies with i⁺≲ 26 detected in a region roughly contiguous with the area covered by the VLA-COSMOS survey. The wavelength range covered by these observations (30 broad, medium, and narrowband filters) extends all the way from the UV at 1550 Å to the MIR at 8 μm. Capak et al. (2007, 2008) provide a complete description of these observations.

Spectroscopic data have been gathered for more than 20,000 sources in the COSMOS field, e.g., by the zCOSMOS survey (Lilly et al. 2007) and Sloan Digital Sky Survey (SDSS; York et al. 2000), or in Magellan/IMACS and Keck/Deimos follow-up observations dedicated to specific (classes of) sources (e.g., Trump et al. 2007, 2009; J. Kartaltepe et al. 2010, in preparation; M. Salvato et al. 2010, in preparation). If a reliable spectroscopic redshift is available it is favored over the photometric redshift estimate. The choice of the best possible distance measurement for our radio and IR sources is described in detail in Appendix A.5.

The XMM-Newton COSMOS Survey (Hasinger et al. 2007; Cappelluti et al. 2007, 2009) has detected a total of 1887 bright (≳2×10⁻¹⁵ erg cm⁻² s⁻¹ in the 0.5–10 keV band) X-ray sources over 90% (1.92 deg²) of the COSMOS field. A large fraction of these are associated with AGN and hence provide a means of identifying AGN-powered radio and IR sources in our sample which is complementary to our primary classification scheme introduced in Section 3. For our subsequent analysis, we rely on a list of XMM-Newton sources (Cappelluti et al. 2009) with unique and secure optical counterparts (see M. Brusa et al. 2010, in preparation) and SED fits to the UV to MIR photometry performed by Salvato et al. (2009).

Due to the differing characteristics (resolution, astrometric accuracy) of the radio and IR data, the determination of counterparts at other wavelengths differed somewhat for the radio- and the IR-selected sample. Which candidate counterparts are incorporated in the final sample and which are rejected is determined by the goals of this study; in our case, it is more important to select objects with the cleanest possible radio and IR flux measurements rather than having a statistically complete sample. The details of the band merging between the IR and the radio catalogs and the subsequent exclusion of ambiguous counterparts are presented in detail in Appendix B. Here we summarize the most important properties of the radio- and IR-selected samples.

2.3.1. The Radio-Selected Sample

Based on the positions of ⩾5 σ 1.4 GHz detections in the VLA-COSMOS Deep Project image, we searched for IR counterparts in the S-COSMOS 24 and 70 μm catalogs which have S/N≳ 3. Counterparts were determined by direct positional matching of radio and IR coordinates with search radii corresponding to approx. FWHM/3 of the IR PSFs of the respective MIPS bands. If no counterpart was found, a 3σ point-source detection limit was determined based on the corresponding uncertainty images. Radio sources with ambiguous IR counterparts—i.e., in the presence of more than one potential counterpart or if the counterpart had not been uniquely assigned to a single radio source—have been excluded from the analysis of the paper. The match with the COSMOS multiwavelength and spectroscopy catalogs provides distance estimates for 73% of the radio-selected sample as well as photometry from the UV to the MIR which is used to separate galaxies dominated by star formation or AGN emission (see Section 3). In the upper panel of Figure 2, we show histograms of the separation between radio source positions and the location of the optical and IR counterparts. Note that the distance is normalized by the width of the broader PSF of the two involved bands. Figure 3 shows the 24 and 70 μm flux distribution of the radio sources, including information on whether the flux constraint is a well-defined measurement or an upper flux limit.

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Figure 4 (left-hand panel) explicitly shows how the fraction of sources that have a directly detected counterpart in either or both of the MIPS bands or only upper flux limits changes as a function of redshift (38% of the redshifts are spectroscopically, 64% photometrically determined).

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2.3.2. The IR-Selected Sample

The IR-selected sample is based on sources listed in the S-COSMOS 24 μm catalog that satisfy the criterion S_ν(24 μm)⩾ 0.3 mJy. This criterion ensures that the IR-selected sample is well matched to the 70 μm and 1.4 GHz data as far as the sampling of the IR luminosity function is concerned. To reduce the likelihood of false identifications due to the positional uncertainty of the 24 μm sources we searched for IRAC counterparts, the positions of which were used as a prior in the subsequent band merging with the other wavelengths. If no IRAC counterpart was available we also admitted unambiguous matches with optical sources. 70 μm counterparts to the 24 μm sources with S/N⩾ 3 were determined and validated following exactly the same approach as in the radio-selected sample. For those 24 μm sources which did not already have a known radio counterpart (determined in the construction of the radio-selected sample) we checked whether they are associated with a counterpart having S/N> 3. All new detections satisfying this criterion were then added to the list of radio counterparts with S/N⩾ 5 that were already known from the construction of the radio-selected sample. 24 μm sources that are undetected at 70 μm and/or 1.4 GHz are assigned 3 σ upper flux bounds.

The distributions of separations between all 24 μm sources and their counterparts in the optical, FIR, and 1.4 GHz maps are given in Figure 2(b). The contribution of flux limits and well-defined flux measurements as a function of flux and S/N at 70 μm and 1.4 GHz is illustrated in Figure 5. Finally, in the right-hand panel of Figure 4 we show at different redshifts which fraction of the IR-selected sample has direct detections or upper flux density limits at 70 μm and/or 1.4 GHz. Spectroscopic or photometric redshift measurements are available for 80% of the objects in the IR-selected sample. The remaining sources are either not bright enough for spectroscopy or have flux information in too few bands to derive a photometric redshift based on SED fitting.

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2.3.3. The Jointly Selected Sample

The jointly selected sample is the union of the radio- and IR-selected samples presented in Sections 2.3.1 and 2.3.2. As such it contains 6863 sources: 1560 sources that are only detected at 1.4 GHz, 3960 sources that are only detected at 24 μm and, finally, 1341 sources which are selected at both wavelengths. In Table 1 we summarize the available redshift information for the jointly radio- and IR-selected sample, as well as separately for the radio- and IR-selected samples.

Table 1. Summary of Sample Properties

Category	No. of Src.	% of Cat. Src.	Remarks
Radio-selected sample
All	2901	100	Initial 1.4 GHz sample
No redshift	748	25.8	Includes 21 1.4 GHz sources outside area covered by multi-λ photometry
Spec-z	769	26.5
Photo-z	1232+114	46.4	Regular + AGN photo-zs
Unreliable redshift	38	1.3
Usable	2020	69.6	No ambiguous counterparts, reliable redshift, and AGN/SF classification available
Pr (SF) ⩾ 0.5	766	26.4	Optically selected SFGs with (u − K)< 2.36 (38% of "usable" sample)
Pr (SF) < 0.5	1254	43.2	Optically selected AGN with (u − K)⩾ 2.36 (62% of "usable" sample)
IR-selected sample
All	5301	100	Initial 24 μm sample restricted to Sν(24 μm)⩾ 0.3 mJy
No redshift	1003	18.9	Includes 304 24 μm sources outside area covered by multi-λ photometry
Spec-z	1254	23.7
Photo-z	2712 + 231	55.5	Regular + AGN photo-zs
Unreliable redshift	101	1.9
Usable	3259	61.5	No ambiguous counterparts, reliable redshift & AGN/SF classification available
Pr (SF) ⩾ 0.5	1822	34.4	Optically selected SFGs with (u − K)< 2.36 (56% of "usable" sample)
Pr (SF) < 0.5	1437	27.1	Optically selected AGN with (u − K)⩾ 2.36 (44% of "usable" sample)
Jointly IR- and radio-selected sample
All	6863	100	1560/3960 sources selected only at 1.4 GHz/24 μm; 1341 in both catalogs
No redshift	1620	23.6	Includes 321 sources outside area covered by multi-λ photometry
Spec-z	1486	21.7
Photo-z	3362 + 275	53.0	Regular + AGN photo-zs
Unreliable redshift	120	1.7
Usable	4454	64.9	No ambiguous counterparts, reliable redshift, and AGN/SF classification available
Pr (SF) ⩾ 0.5	2215	32.3	Optically selected SFGs with (u − K)< 2.36 (49.7% of "usable" sample)
Pr (SF) < 0.5	2239	32.6	Optically selected AGN with (u − K)⩾ 2.36 (50.3% of "usable" sample)

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In Figure 6, the IR and radio fluxes of our sources are compared. The color coding of the data points distinguishes three kinds of sources: in black those which have entered both the 1.4 GHz catalog as well as the 24 μm catalog (restricted to sources with flux density larger than 0.3 mJy), in green 1.4 GHz sources without counterpart in the 24 μm catalog, and in orange those 24 μm detected sources which do not have a counterpart in the VLA-COSMOS Joint catalog. The sources from these three different categories have been added to the plot in random order to prevent the symbols of the initially plotted category from being systematically hidden by the successively overplotted data in common regions of flux space. Figure 6(c) confronts the fluxes in the two selection bands; the empty rectangle in the lower left corner of this panel reflects the selection criteria at 1.4 GHz and 24 μm. Since the 24 μm catalog is flux limited, essentially all upper 24 μm flux limits lie at or below the critical flux threshold; upper 1.4 GHz flux limits for undetected 24 μm sources on the other hand are also encountered at higher 1.4 GHz flux values than the sharp cutoff at S_ν(1.4 GHz)≈ 0.05 mJy because the radio catalog was constructed using a S/N criterion. Note that the region where both the 1.4 GHz and 24 μm flux density clearly exceed the respective selection thresholds contains some sources which are not included in both the catalog of 24 μm and that of 1.4 GHz detections (see orange and green symbols in the area where S_ν(1.4 GHz)> 0.1 mJy and S_ν(24 μm)> 0.3 mJy). Two reasons can be responsible for this: (1) minor incompleteness of the catalogs or (2) spatial variations in the background noise which, at a given flux, lead to certain sources not being detected at the significance level required for inclusion in the original source list.

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The selection effects that are the topic of this section arise in flux-limited samples when flux information from one of the selection bands is directly used in the computation of the quantity being studied. In the present case the critical quantity is the logarithmic IR/radio flux ratio q, but analogous effects need to be considered in the context of studies of the distribution of spectral indices at different radio frequencies (e.g., Kellermann 1964; Condon 1984), of X-ray to optical continuum slopes of AGN (Francis 1993), or of the M_•–σ and M_•–L relationships (Lauer et al. 2007).

In Figure 10, we illustrate the origin of the selection effect: consider the left-hand panel in which the IR-to-radio SEDs of three sources with different observed bolometric flux are distributed along the vertical axis. Each of these three SEDs splits into three branches at the peak of the SED, thereby schematically reflecting the range of observed IR/radio ratios (from top to bottom: 3 σ radio-excess outlier, dashed line; average source, solid line; and 3 σ IR-excess outlier, dotted line). If we impose the indicated selection threshold at 1.4 GHz (red line) the resulting sample will contain (1) all sources of the brightest flux class, regardless of their IR/radio ratio; (2) the source with an average IR/radio ratio and the radio-excess source from objects of the intermediate flux class and; (3) in the faintest flux bin only the radio-excess sources. Since the fainter sources are more abundant (as parameterized by the slope of the differential source counts dN/dS ∝ S^−β, with β> 0) this results in a surplus of radio-excess sources and consequently a low average IR/radio ratio in a radio-selected sample. The right-hand side of Figure 10 shows that an IR-selected sample is biased in the opposite direction, i.e., toward high IR/radio ratios.

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The analytical expression for the difference between the average IR/radio ratio of IR- and radio-selected samples is (Kellermann 1964; Condon 1984; Francis 1993; Lauer et al. 2007)

$$\begin{equation} \Delta q_{\rm bias} = {\rm ln}(10)\,(\beta -1)\,\sigma _q^2. \end{equation} \tag{ 5 }$$

It thus depends on β, the power-law index of the source counts, and on σ_q, which is the dispersion of the IR/radio relation. Note that this offset will occur regardless of the relative depth of the two involved bands. An estimate of the "intrinsic" (i.e., unbiased) IR/radio ratio can be obtained by constructing the sample using an unrelated selection criterion like optical luminosity, mass, or morphological type (Lauer et al. 2007).

Since the recent work on the evolution of the IR–radio relation at intermediate and high redshift was often based on flux-limited surveys, we would expect most of the findings to be affected by this selection bias to a certain extent. In Table 2, we have collected the selection criteria and average values of q (final column) that were published in the literature during the last decade.

We see that broadly speaking the various IR/radio diagnostics have values q₂₄≈ 1, q₇₀≈ 2.1, q_FIR≈ 2.3, and q_TIR≈ 2.6. These different values are not the result of selection effects but reflect if the IR filter covers a wavelength range that is close to the IR SED peak or a part of the SED with lower energy content. In the following paragraph, we will discuss the plausible influence of selection effects on the various measurements of q₂₄ and q_FIR, in particular.

Due to the high sensitivity of the MIPS 24 μm band, many of the papers listed in Table 2 have studied the IR/radio ratio q₂₄. The radio-selected samples of Appleton et al. (2004) and Ibar et al. (2008) find that q₂₄∈ [0.94, 1], depending on the choice of the IR template used for the K correction. The local IR-selected sample of Rieke et al. (2009) on the other hand has a mean q₂₄ of ∼1.25 and shows some signs of variations with IR luminosity. The offset between the means of the radio-selected samples and the IR-selected data set is ∼0.3 dex, in good agreement with the predicted Δq_bias = 0.31 of Equation (5) if we set σ_q≈ 0.3 in accordance with observations (e.g., Yun et al. 2001; Bell 2003; Appleton et al. 2004; Ibar et al. 2008) and under the simplified assumption of Euclidean source counts (β = 2.5). In the case of the FIR/radio flux ratio q_FIR we can compare the two radio-selected (sub)samples of Garrett (2002) and Sajina et al. (2008) that have q_FIR≈ 2 with a jointly radio- and submillimeter-selected mean of 2.07 from Kovács et al. (2006) and mean values q_FIR∈ [2.2, 2.4] for IR-selected (Younger et al. 2009; Sajina et al. 2008) or essentially volume-limited samples in Bell (2003) and Yun et al. (2001). As with q₂₄ there is thus evidence of a ∼0.3 dex shift in q_TIR between radio-selected and other samples (but see also Swinbank et al. (2008) for an alternative explanation of the IR/radio ratios measured by Kovács et al. (2006)). As far as we know no measurement of q₇₀ in an IR-selected sample exists but the compilation in Table 2 shows that reassuringly all determinations of q₇₀ based on radio-selected samples (Appleton et al. 2004; Frayer et al. 2006; Seymour et al. 2009) are quite similar.

The radio stacking experiments of Boyle et al. (2007), Beswick et al. (2008), and Garn et al. (2009) do not fit the picture which is probably due to the different nature of the analysis. Nevertheless, it cannot be excluded that part of the variations in the other studies are due to field-to-field variance or different assumptions about IR SEDs and the radio spectral slope. To this end, we will test in Section 6 whether or not the offset between our IR- and radio-selected samples—that have been consistently constructed from the same parent data sets—conforms to our expectation. If so, it would be strong support for selection effects alone being able to reconcile the seemingly discrepant measurements of average IR/radio properties in the literature.

Discarding the information from undetected counterparts introduces a second source of bias in addition to the selection effects mentioned in Section 5.1. It arises from the unrepresentative sampling of the true distribution function of IR/radio ratios by sources which are directly detected in all involved bands. We would like to emphasize that the shift Δq_bias in Equation (5) is the difference between the mean q of IR- and radio-selected samples with correctly sampled distribution functions. Δq_bias cannot be compensated by accounting for upper or lower limits on q due to undetected IR or radio counterparts in the two different samples; as discussed in the previous section, the mean IR/radio ratio measured in an IR- and radio-selected sample only brackets the value one would measure with an unbiased data set which we can best approximate by a sample jointly selected at IR and radio wavelengths (see Section 2.3.3).

The ratio of two flux constraints that could be either a well-defined measurement or an upper limit will render an upper or lower bound, a well-defined value, or be indeterminate (if both numerator and denominator are limits). Since we use the pooled information from a radio- and IR-selected sample in this study the latter case never occurs. We do expect, however, to encounter upper limits on IR/radio ratios from radio-selected sources that are not detected in the IR or lower limits if the radio counterpart of an IR-selected source was too faint to be detected (see Appendices A.1 and A.2).

Let q_i (i = 1, ..., n) be the actual value of the flux ratio for each of the n sources in a suitably defined sample (e.g., the population in a certain slice of redshift). As a consequence of the noise characteristics in the radio and IR images, q_i can only be measured if it lies in the interval [q⁻_i, q⁺_i], where q⁻_i and q⁺_i are the upper and lower limits on the flux ratio, respectively. These limits may be different for each source. Our knowledge about the distribution of IR/radio ratios after carrying out all our measurements can thus be summarized with two vectors of variables, ${\mathcal {\bm Q}}$ and δ:

$$\begin{equation} \begin{array}{c} \dsty \mathcal {Q}_i = \hbox{ max(min($q_i$, $q^+_i$), $q^-_i$)} \\ \ms \dsty \delta _i =\left\lbrace \begin{array}{@{}l@{\quad }l} -1 & \mbox{if }q_i<q^-_i\\ \ms 0 & \mbox{if }q_i \in [q^-_i, q^+_i]\\ \ms 1 & \mbox{if }q_i>q^+_i. \end{array}\right. \end{array} \end{equation} \tag{ 6 }$$

In survival or life time analysis, the action of imposing measurement limits is referred to as "censoring." A variable $\mathcal {Q}_i$ is said to be left censored if $\mathcal {Q}_i<q^-_i$ and right censored if $\mathcal {Q}_i>q^+_i$. If both kinds of censoring occur in a data set it is called doubly censored, otherwise one talks of single censoring. During the remainder of the paper we will use the terms "limit" and "censored measurement" interchangeably.

In Appendix C, we sketch the steps that are involved in going from the information (${\mathcal {\bm Q}}$, δ) to the distribution function of the IR/radio ratios. Inferring the true distribution of the q_i of a sample is essential for the calculation of its average IR/radio properties. In Section 6, we will construct distribution functions for data sets that are both singly and doubly censored. Recipes for dealing with the former case are plentiful in texts on survival analysis (see, e.g., Feigelson & Nelson (1985) for applications to astronomy) such that we only include some brief remarks in Section C.2. Since the more general case of double censoring is not as widely used in astronomical applications, the most important formulae and useful computational guidelines are provided in Appendix C.1.

The methods described in Appendix C have been implemented using Perl/PDL²² scripts written by M.T.S. Their correct functionality was verified with examples in the literature. In particular, we checked that our implementation of the algorithm for the calculation of the doubly censored distribution function (Schmitt 1985)—when applied to the special case of singly censored data—gave the same results as the scripts based on the Kaplan–Meier product limit estimator (Kaplan & Meier 1958; see also Appendix C.2).

6.1.1. Observed Flux Ratios

The observed 24 μm/1.4 GHz flux ratio q_24, obs is plotted against redshift in Figure 11 for SFGs (top) and AGNs (bottom). Sources are assigned to the two categories depending on whether Pr (SF) is larger or smaller than 50% (see Section 3).

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While there clearly are many radio-loud sources in our AGN sample, Figure 11(b) shows that a majority of the objects assigned to the AGN category display very similar IR/radio ratios as the SFGs. We will discuss this observation in more detail in Section 6.3. At the same time, the sample of SFGs also includes a number of radio-excess sources. They usually have photometric redshift estimates and mostly lie at 1 <z< 3. This roughly corresponds to the redshift range in which photometrically determined redshifts are subject to the largest uncertainty because the 4000 Å break is only sampled by broad and widely spaced photometric bands. As a consequence, absorption features and emission lines from AGN and SF systems often interfere with each other in the same filter. Even though we did attempt to remove all unreliable redshifts—as described in Appendix A.5—it thus seems probable that at least some of these cases are due to wrong distance estimates and hence to the selection of an inappropriate optical SED. Since this results in a faulty (u−K) color, the source in question could then have been assigned to the SF rather than the AGN category. Another possibility is that the peak of AGN activity at z∼ 2 (e.g., Wolf et al. 2003; Richards et al. 2006) also influences the SF sample due to the statistical nature of the identification of SFGs and because especially AGN in composite systems could have been classified as star forming. Finally, we tried to assess if unobscured Type 1 AGNs represent a significant fraction of the nominally SF radio excess sources in the pertinent redshift range. Based on the confidence class (see Appendix A.5.1) of those objects for which follow-up spectroscopy was available, we estimate that only ∼5% are quasars classified as SF due to their blue color.

The mean value of q_24. obs decreases as a function of redshift. We will show later on (see Figure 20) that this decrease agrees with the variations local LIRGs (detectable only out to z∼ 1; see vertical dotted lines in panel (a)) and ULIRGs would display if redshifted.

In Figure 12, we plot the observed 70 μm/1.4 GHz flux ratio q_70. obs of our sources as a function of redshift. All symbols and colors are exactly as in the previous figure. Note that censored measurements due to flux limits at 70 μm are more frequent than was the case for q_24. obs because the 70 μm observations are much shallower. The observed flux ratio before K correction shows the same decline at higher redshifts as was seen for the observed 24 μm/1.4 GHz flux ratios.

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6.1.2. Evolution of q₂₄ and q₇₀ with Redshift

As described in Section 4, all sources classified as SF were fit with IR SEDs in order to derive IR luminosities. As a by-product of the template fitting we can immediately obtain rest-frame (i.e., K corrected) 24 and 70 μm flux densities by convolving the best-fitting SED with the filter response functions of MIPS. In the following, we define the K-corrected 24 and 70 μm fluxes as the average of the values obtained from the libraries of Chary & Elbaz (2001) and Dale & Helou (2002) and use them to construct monochromatic rest-frame flux density ratios q_24, 0 and q_70, 0. The associated 1.4 GHz flux densities have been K corrected according to Equation (8).

To quantify the evolution of q_24, 0 in the joint IR- and radio-selected sample, we

• 1.  
  Bin the data such that each redshift slice contains an identical number of objects (∼250). The number of bins is kept limited to guarantee that the distribution function of q_24, 0 is sampled sufficiently.
• 2.  
  Run the iterative algorithm outlined in Appendix C.1 to find the cumulative distribution function of q_24, 0 at each redshift. The median immediately follows from this computation as does the scatter in the population which we obtain by fitting a Gaussian distribution with known mean (equal to the previously determined median) to the distribution function. The choice of the Gaussian is motivated by the shape of the local IR–radio relation (Helou et al. 1985; Yun et al. 2001; Bell 2003).
• 3.  
  Determine the evolution of the average IR/radio ratio by fitting a linear trend line to the medians. Only measurements at z⩽ 1.4 are considered for this since the scatter at higher redshifts is found to increase abruptly, thus making the determination of the median uncertain.

Steps 1 and 2 are also carried out individually for the sample of IR- and radio-selected galaxies. The results are shown in Figure 13. Since the cumulative distribution function is normalized it lies in the range between zero and unity and can thus be regarded as the probability of obtaining a measurement of q_24, 0 which is less than—in the case of the radio-selected sample (light gray curve)—or in excess of—for the IR-selected sample (dark gray curve)—the ordinate. The distribution function of the doubly censored union of the IR- and radio-selected samples is also parameterized such that it runs from 0 to 1 with increasing q_24, 0. It is plotted in black together with a dashed red line which shows the corresponding best-fitting Gaussian distribution. The intersection of the black curve with the 50% probability line (dotted horizontal line) defines the median value of q_24, 0.

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Figure 13 demonstrates that the median of the radio-selected population lies systematically below that of the IR-selected objects. The shift is approximately 0.35 dex at low redshift and grows to about 0.7 dex beyond z∼ 1. The increase is probably caused by the intrinsically higher scatter in the IR–radio relation at high luminosities (Yun et al. 2001; Bressan et al. 2002), possibly in combination with the reduced reliability of photometric redshifts and/or some falsely classified AGN that begin to affect the sample starting at z∼ 1.3. A shift of ∼0.35 dex as observed at z< 1 where the accuracy of our measurements is highest agrees fairly well with the prediction of Equation (5) and is hence a likely explanation for differences between previously reported average IR/radio properties of both local and high-z galaxies (e.g., Appleton et al. 2004; Ibar et al. 2008; Rieke et al. 2009).

In the lower panel of Figure 13, we plot the medians 〈q_24, 0〉 of the jointly IR- and radio-selected SFGs (black dots) at different redshifts on top of the K-corrected values q_24, 0 (colors and symbols are identical to those in Figure 11(a)). The error bars mark the 95% confidence interval associated with the median. Table 3 lists the median and scatter of q_24, 0 which were determined with survival analysis in each of the redshift bins of Figure 13. In addition to the measurements carried out on the jointly IR- and radio-selected sample the table also provides the according values for the IR- and radio-selected samples individually.

Table 3. Median of K-corrected Logarithmic 24 μm/1.4 GHz Flux Ratios, q_24, 0, as a Function of Redshift (see also Figure 13)

Redshift	〈q24, 0〉	〈q24, 0〉	〈q24, 0〉	$\sigma _{q_{24}}$
	Radio-selected sample	IR-selected sample	Jointly selected sample
0.08 ⩽z< 0.23	1.02+0.08−0.07	1.35+0.08−0.12	1.31+0.10−0.05	0.37 ± 0.03
0.23 ⩽z< 0.33	0.88+0.16−0.10	1.28+0.07−0.07	1.26+0.07−0.06	0.25 ± 0.03
0.33 ⩽z< 0.45	0.94+0.11−0.04	1.29+0.04−0.08	1.27+0.08−0.04	0.31 ± 0.04
0.45 ⩽z< 0.67	0.97+0.08−0.14	1.26+0.05−0.14	1.26+0.02−0.08	0.35 ± 0.03
0.67 ⩽z< 0.82	0.83+0.18−0.16	1.29+0.06−0.06	1.23+0.06−0.03	0.36 ± 0.03
0.82 ⩽z< 0.94	0.78+0.20−0.17	1.32+0.03−0.09	1.31+0.03−0.06	0.35 ± 0.03
0.94 ⩽z< 1.14	0.77+0.18−0.26	1.36+0.07−0.10	1.29+0.07−0.09	0.49 ± 0.04
1.14 ⩽z< 1.51	0.75+0.12−0.15	1.47+0.11−0.07	1.16+0.09−0.10	0.91 ± 0.06
1.51 ⩽z< 2.00	0.96+0.09−0.21	1.63+0.12−1.01	1.35+0.10−0.06	0.79 ± 0.04
2.00 ⩽z< 4.50	1.06+0.11−0.28	1.93+0.11−0.21	1.62+0.10−0.12	0.88 ± 0.06

Notes. The average IR/radio properties are given for three different samples of SFGs: a radio-selected sample, an IR-selected sample, and the combination of the two of these (from left to right). For the jointly selected sample, we also give the scatter in the relation (right-most column). All errors are 2 σ uncertainties.

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Using the 2 σ errors on the medians as weights we fit them with a model of linear redshift evolution. The best-fitting trend of 〈q_24, 0〉 versus z is shown in black in the lower window of Figure 13 (see Table 4 for the parameterization of the line). Because the fit was carried out with respect to linear redshift space while the plot has a logarithmically scaled redshift axis it is curved. Within the errors the slope d〈q_24, 0〉/dz = −0.015 ± 0.136 is consistent with no evolution of the K-corrected 24 μm/1.4 GHz ratio at z⩽ 1.4 (the maximal distance out to which the precision of the photometric redshifts is high). The y-axis intercept of the trend line at z = 0 is in agreement with the recent analysis of Rieke et al. (2009) who find that q_24, 0 = 1.22 with a scatter of 0.24. In our sample we find 〈q_24, 0〉_z=0= 1.28 ± 0.10 (where the error states the formal 1 σ uncertainty from the linear fit). For a comparison between the average IR/radio properties of radio-selected samples, we can refer to the studies of Appleton et al. (2004) and Ibar et al. (2008); they report an average q_24, 0 = 0.94–1, depending on the IR-SED adopted to K correct to the rest frame. These values agree well with the range of medians 〈q_24, 0〉∈ [0.8, 1] measured for radio-selected COSMOS data at intermediate redshift (see the left-most column of Table 3).

Our convention for choosing SFGs states that Pr (SF) must be at least 50%. In Figure 14, we assess how changing this threshold affects the redshift evolution that is inferred from the data. The variation of the parameters of the best-fitting evolutionary trend line is shown in the upper- and lower-most window of Figure 14 (y-axis intercept and slope, respectively). A black symbol in the middle of the displayed data range marks the results that were shown in Figure 13. They are fully consistent with the evolution found if a more conservative threshold—e.g., at Pr (SF) = 66%—for the selection had been chosen. It is interesting that the inclusion of a significant fraction of sources with a probability of up to 80% of being AGN does not alter the results either. This is a strong indication that our sample of optically selected AGN contains many objects with IR/radio properties that closely resemble those of SF systems. Similar observations were made by, e.g., Sopp & Alexander (1991) and Roy et al. (1998), who studied local samples of radio-quiet quasars and/or Seyfert 1 sources lacking a compact nucleus. The middle row of Figure 14 shows that while the average values of q_24, 0 are similar for many SFGs and AGNs the latter are subject to a larger scatter as was previously found by, e.g., Condon et al. (1982), Obrić et al. (2006), and Mauch & Sadler (2007).

As an additional test of the robustness of our findings, we checked if the evolutionary trend in SF samples selected through (u−K) or P1 differs. For the radio-selected sample where both colors were available we found equivalent results regardless of the chosen approach.

Figures 15 and 16 (the results of which are summarized in Table 5) repeat the analysis of Figures 13 and 14 for the K-corrected 70 μm/1.4 GHz flux ratio. Note that in comparison with Figure 12 the number of censored measurements is much smaller when we consider rest-frame IR/radio ratios q_70, 0 rather than observed flux ratios q_70, obs. The reason is that we do not require a 70 μm detection for the IR template fitting but also fit objects which have a limit at 70 μm and a direct detection at 24 μm.

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The plot of q_70, 0 versus redshift (lower panel of Figure 15) as well as Figure 16, which illustrates the stability of the findings with respect to changes in the selection criterion for SFGs, show that q_70, 0 behaves in a similar way as was found for q_24, 0. The extrapolated average 70 μm/1.4 GHz flux ratio at z = 0, 〈q_70, 0〉_z=0 equals 2.31 ± 0.09 and if AGNs are included the scatter in the relation increases in analogy to what was found for q_24, 0. As for q₂₄, the evolutionary slope d〈q_70, 0〉/dz = −0.123 ± 0.135 (slope and normalization of the evolutionary trend line are logged in Table 4) is consistent with zero.

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Table 4. Slope (Column 1) and y-axis Intercept (Column 2) of the Best-fitting Linear Trend to the Evolution of the Three Investigated IR/radio Parameters q_24, 0, q_70, 0, and q_TIR

IR/Radio Parameter	d〈q〉/dz|z<1.4	〈q〉z=0	〈σq〉|z<1.4
q24, 0	−0.015 ± 0.136	1.275 ± 0.098	0.417 ± 0.038
q70, 0	−0.123 ± 0.135	2.314 ± 0.091	0.392 ± 0.035
qTIR	−0.268 ± 0.115	2.754 ± 0.074	0.412 ± 0.037

Note. The linear fit was performed using the median logarithmic IR/radio ratios in all redshift slices at z< 1.4. The third column gives the average scatter in the relation measured over this redshift range. The states errors are 1 σ uncertainties.

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The average IR/radio properties of the radio-selected samples of Appleton et al. (2004) and Frayer et al. (2006)²³ are 2.16 ± 0.17 and 2.10 ± 0.16, respectively. Although the agreement with our findings is not quite as good as in the case of q₂₄, they are nevertheless consistent (within both the formal error and the scatter) with the range of medians 〈q_70, 0〉∈ [1.7, 2.1] at intermediate redshift in our radio-selected sample (see Table 5). To our knowledge there so far has been no comparable study which uses an IR-selected sample to compute an average q₇₀.

Table 5. As for Table 3 but for the K-corrected Logarithmic 70 μm/1.4 GHz Flux Ratio q_70, 0

Redshift	〈q70, 0〉	〈q70, 0〉	〈q70, 0〉	$\sigma _{q_{70}}$
	Radio-selected sample	IR-selected sample	Jointly selected sample
0.08 ⩽z< 0.23	2.07+0.06−0.09	2.41+0.09−0.11	2.39+0.08−0.08	0.39 ± 0.04
0.23 ⩽z< 0.33	1.91+0.16−0.10	2.27+0.03−0.05	2.26+0.06−0.04	0.25 ± 0.03
0.33 ⩽z< 0.45	1.99+0.10−0.08	2.30+0.05−0.08	2.28+0.07−0.06	0.30 ± 0.04
0.45 ⩽z< 0.67	1.94+0.10−0.15	2.23+0.03−0.08	2.24+0.02−0.07	0.36 ± 0.03
0.67 ⩽z< 0.82	1.77+0.14−0.13	2.23+0.04−0.08	2.20+0.02−0.05	0.39 ± 0.03
0.82 ⩽z< 0.94	1.79+0.18−0.20	2.29+0.07−0.05	2.22+0.06−0.02	0.28 ± 0.03
0.94 ⩽z< 1.14	1.67+0.26−0.15	2.32+0.06−0.10	2.25+0.06−0.10	0.46 ± 0.04
1.14 ⩽z< 1.51	1.67+0.12−0.14	2.33+0.08−0.13	2.02+0.14−0.11	0.83 ± 0.05
1.51 ⩽z< 2.00	1.78+0.10−0.22	2.45+0.12−1.66	2.17+0.08−0.06	0.74 ± 0.04
2.00 ⩽z< 4.50	1.85+0.10−0.28	2.71+0.07−0.18	2.38+0.11−0.08	0.89 ± 0.06

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In the local universe, the correlation of IR and radio flux is tightest if integrated (F)IR luminosities rather than monochromatic flux ratios are considered. To complement the analysis of Section 6.1, we thus show in this section the correlation of TIR (8–1000 μm) and 1.4 GHz luminosity as parameterized by the TIR/radio ratio q_TIR for our VLA- and S-COSMOS data.

The computation of the distribution functions for the parameter q_TIR is carried out following the same steps described in Section 6.1.2. The results are shown for a number of redshift bins in Figure 17 where we also compare the median derived for the jointly IR- and radio-selected SFGs with the local value of 〈q_TIR〉_z=0 = 2.64 ± 0.02 (Bell 2003; vertical dashed line). Our average values 〈q_TIR〉 in the range z< 1.4 (see Table 6) lie to either side and always remain well within the dispersion of the local measurement of Bell (2003).

Table 6. As for Tables 3 and 5 but for the Logarithmic TIR/1.4 GHz Flux Ratio q_TIR

Redshift	〈qTIR〉	〈qTIR〉	〈qTIR〉	$\sigma _{q_{\rm TIR}}$
	Radio-selected sample	IR-selected sample	Jointly selected sample
0.08 ⩽z< 0.23	2.41+0.10−0.12	2.75+0.05−0.11	2.76+0.02−0.08	0.39 ± 0.03
0.23 ⩽z< 0.33	2.29+0.23−0.09	2.63+0.02−0.07	2.64+0.05−0.04	0.29 ± 0.04
0.33 ⩽z< 0.45	2.37+0.10−0.16	2.71+0.04−0.07	2.70+0.05−0.06	0.34 ± 0.04
0.45 ⩽z< 0.67	2.28+0.09−0.12	2.61+0.07−0.12	2.56+0.05−0.08	0.39 ± 0.04
0.67 ⩽z< 0.82	2.09+0.12−0.14	2.54+0.04−0.08	2.51+0.03−0.05	0.39 ± 0.03
0.82 ⩽z< 0.94	2.07+0.20−0.16	2.56+0.04−0.07	2.51+0.06−0.02	0.28 ± 0.03
0.94 ⩽z< 1.14	1.98+0.23−0.16	2.61+0.06−0.08	2.56+0.06−0.12	0.48 ± 0.05
1.14 ⩽z< 1.51	1.97+0.10−0.15	2.64+0.09−0.11	2.29+0.16−0.08	0.78 ± 0.05
1.51 ⩽z< 2.00	2.10+0.09−0.22	2.77+0.12−0.26	2.49+0.08−0.06	0.75 ± 0.04
2.00 ⩽z< 4.50	2.17+0.10−0.28	3.04+0.08−0.19	2.72+0.10−0.10	0.85 ± 0.06

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The evolution of q_TIR is shown in the lower panel of Figure 17 using the same presentation of the data as for the monochromatic IR/radio flux ratios. Since the latter were derived based on the IR templates which are used here to calculate the integrated IR luminosity, we expect by construction that the evolutionary trend is in good qualitative agreement with the findings of Section 6.1. For the same reason we cannot expect to observe a reduced scatter in the values of q_TIR with respect to those of the monochromatic flux ratios as the spread in the properties of the best-fitting IR SEDs must manifest itself in q_24, 0 and q_70, 0 as well.

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The line parameters for the evolution of 〈q_TIR〉 are given together with those of 〈q_24, 0〉 and 〈q_70, 0〉 in Table 4: in contrast to q_24, 0 and q_70, 0 the best-fitting evolutionary trend for 〈q_TIR〉 suggests a decrease of the average TIR/radio ratio by 0.35 dex out to z∼ 1.4. However, this slope is detected at the 2 σ significance level and predicts a median 〈q_TIR〉 at z∼ 1.4 that still lies within the dispersion measured in our lowest redshift bin. It thus seems unlikely that the evolutionary signal is real, especially in view of the results of Section 6.5 where we measure an average 〈q_TIR〉_z>2.5 that is in excellent agreement with the local value for a subset of highly redshifted galaxies in the COSMOS field. An examination of the evolutionary slopes for 〈q_24, 0〉, 〈q_70, 0〉, and 〈q_TIR〉 in Table 4 shows that they become more negative along this sequence. This could be related to a number of radio-excess sources with 1 ≲z≲ 3 which are part of our optically selected SF sample (visible as a diffuse cloud of upper limits and detections below the main locus of symbols in all our plots of q versus z; see also our comment in Section 6.1.1) and that tend to lower the average IR/radio in this redshift range. If these objects were falsely classified composite sources or AGNs, the increased emission at 24 μm from their hot dust might be able to compensate the radio excess, thus leading to zero evolution in q_24, 0 as observed. q_70, 0 and q_TIR, on the other hand, sample mainly IR light from star formation and hence are lowered in the presence of excess radio emission. This scenario can also explain why the evolutionary slope of 〈q_24, 0〉 is insensitive to the selection criterion for SFGs (see Figure 13), while it varies in the same sense as described above in the case of 〈q_70, 0〉 and 〈q_TIR〉.

The analysis of the previous sections revealed (see Figures 14, 16, and 18; also Figures 11 and 12) that the IR/radio properties of SFGs are shared by many of the AGN-bearing systems in our sample. In this section, we will study this in more detail. We first test (Section 6.3.1) if it remains valid for a subsample of sources which are detected in X-rays and have been found to host an AGN using a different approach than the classification scheme introduced in Section 3. In Section 6.3.2, we then compute (in different redshift bins at z≲ 1–1.4) the relative frequency of AGN and SF sources as a function of the IR/radio ratio.

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6.3.1. IR/Radio Properties of X-ray Detections

At the sensitivity of the XMM-Newton observations of the COSMOS field a large fraction of the detected sources is expected to be powered by AGN. This is confirmed by Salvato et al. (2009) who have shown that ∼70% of the XMM-Newton sources have UV to NIR SEDs which contain an AGN component. In Figure 19 we compare the observed 24 μm and 70 μm to radio flux density ratios q_24, obs and q_70, obs of X-ray detected AGN hosts at different redshifts with the predicted IR/radio properties of model SFGs (colored tracks²⁴; see footnote and the text of Section 6.3.2 for additional details). Note that according to the analysis of Salvato et al. (2009) the AGN contribution to the UV-NIR SED exceeds 50% for most of these sources. From Figure 19 it is obvious that a majority of the XMM-Newton sources have IR/radio ratios that are perfectly consistent with those expected for starbursts. They are genuine examples of active galaxies in which the AGN, although significantly contributing to the SED at optical and X-ray wavelengths, does not cause significant excess radio emission. Figure 19 therefore is strong evidence that the findings of Sections 6.1 and 6.2 cannot be ascribed to an inadequacy of the method we adopted to distinguish between AGNs and SFGs.

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6.3.2. The Relative Abundance of AGNs and SFGs on the Star-Forming Locus

We first define the "main" locus of SFGs in a plot of observed IR/radio ratio versus redshift. Working with observed flux densities is necessary because we want to avoid imposing a template fit with the IR SED of an SFG on an AGN-bearing source even if it quite probably shares similar IR/radio properties. At each redshift the star-forming locus is centered on the average value—〈q_{24 [70], template}〉—of q_24, obs as predicted by the model SEDs of sources in the observable range of IR luminosities. We then consider a region between +2 σ_q and −2 σ_q around 〈q_{24 [70], template}〉 in which we chart the relative frequency f_AGN(q, z) of AGN. The analysis is restricted to this ±2 σ band because beyond it the sparse sampling of the distribution function of the IR/radio ratios leads to unwanted fluctuations of f_AGN(q, z). The value of σ_q is a representative average of the scatter in our data for the SF population at z≲ 1, i.e., 0.35 dex.

In the upper panel of Figure 20 (Figure 21 shows the same information for q₇₀), we show the expected variations in q_24, obs of high-z galaxies assuming that their SEDs at IR and radio wavelengths are similar to those of local SFGs. SEDs from three different template libraries—as well as that of the starburst M82—are shown for different L_TIR. The tracks are normalized at z = 0 using the best-fit evolutionary trend line displayed in Figure 13. In the background we re-plot (see Figure 11(a)) the observed 24 μm/1.4 GHz ratios of our sample of SF galaxies in order to show how they nicely follow the tracks of the local SEDs. The solid black lines delineate the ±2σ band centered on 〈q_{24, template}〉. The jumps at z∼ 0.5 and 1 occur because the averaging of the IR templates is performed with a discrete and restricted set of IR templates. Since we merely use these boundaries to define the parameter space for the subsequent analysis the discontinuities are inconsequential.

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The expression for the relative AGN abundance which accounts for censored measurements and the use of discrete probability bins is (see derivation in Appendix D)

$$\begin{equation} {{\rm f}_{{\rm AGN}}}(q, z) = \frac{\sum _{i=1}^{n}\,N_{i,\, {\rm eff.}}\times \langle {\rm 1-Pr\,(SF)}\rangle _i}{\sum _{i=1}^{n}\,N_{i,\, {\rm eff.}}\times \langle {\rm Pr\,(SF)}\rangle _i}. \end{equation} \tag{ 10 }$$

Here the summation with respect to i extends over a finite number n of probability bins. In the case of q₂₄ we grouped sources into bins of width ΔPr (SF) = 0.1 in order to have a sufficient number of measurements, and thus to ensure a well-behaved estimate of the distribution function f(q) (computed according to Equations (C1) and (C2)) in each probability bin.

In the lower panel of Figure 20, we present the function f^24 μm_AGN(q, z) in four redshift bins covering z≲ 1.4. The zero-point of the x-axis has been renormalized to the average q₂₄ of the IR templates at the center of the redshift slice. A value of 2 (0.5) on the y-axis implies that at a given value of q₂₄ the relative abundance of AGN and SF systems is 2:1 (1:2). Within the errors f^24 μm_AGN(q, z) is consistent with being unity across the whole width of the star-forming locus at all redshifts. Deviations from the generally smooth variations of f^24 μm_AGN(q, z) can occur on the edge of the assessed range of q₂₄ due to fluctuations caused by poor statistics. There is weak evidence for a gradual decrease of the AGN fraction from about 2/3 to roughly 1/3 as one goes from the region which hosts sources with radio-excess to that populated by sources with excess IR emission. This trend is barely significant but interestingly enough it tilts in the opposite direction as would be expected if, e.g., AGN activity were to manifest itself by exciting increased hot dust emission in the MIR. (Note that in general the radio emission could also be altered by the presence of an AGN, thus making the observed slope less easily interpretable. However, the fact that the distribution of f^70 μm_AGN—which has a radio contribution that is identical to that in f^24 μm_AGN—is essentially flat, suggests that the radio emission is not strongly affected by the AGN.)

The calculation of f^70 μm_AGN(q, z) involved slightly wider probability bins of width 1/8 (to ensure convergence of the distribution function) and was limited to z≲ 1.1 due to the ubiquity of 70 μm non-detections at higher redshift (see the upper panel of Figure 21). f^70 μm_AGN(q, z) appears to be a constant function of q_24, obs with no traces of being tilted as detected with marginal significance for f^24 μm_AGN(q, z), except maybe in the redshift bin z∈ [0.72, 1.06]. Overall, we can thus deduce that our optically selected AGN and SFGs occupy the SF locus in very similar proportions. A possible explanation for this is that both the IR and radio emission are predominantly powered by star formation rather than AGN activity. It is also conceivable, however, that other (combinations of) astrophysical processes conspire to place AGN hosts close to the IR–radio relation (e.g., Sanders et al. 1989; Colina & Perez-Olea 1995).

In a recent work on local IR galaxies, Rieke et al. (2009) have found evidence of variations in the K-corrected average 24 μm/1.4 GHz flux ratio q_24, 0 with IR luminosity L_TIR. According to their analysis q_24, 0 is a constant function of luminosity at L_TIR ⩽ 10¹¹ L_☉ and then begins to rise with increasing luminosity. Using the NVSS- and IRAS-detected SDSS galaxies, I. Morić et al. (2010, in preparation) see an opposite trend of decreasing FIR/radio ratio when they examine q_FIR versus L_1.4 GHz for various types of active galaxies (both star forming and AGN bearing).

We investigate whether or not the SF sources in our sample show any evidence of variations of q_TIR with IR or radio luminosity. Since our K-corrected monochromatic IR/radio ratios are based on the best-fitting TIR SEDs, all luminosity-dependent trends they display will be qualitatively identical to those measured for q_TIR. Comparisons with previous studies are therefore possible even if these used a different IR/radio parameter.

Note that the fact that we are plotting q_TIR against luminosity implies that upper and lower limits cannot always be unambiguously placed along the ordinate. An example is the radio-selected sources in the upper panel of Figure 22 of which we merely know that they must lie to the lower left of their limits. They are indicated by an arrow pointing diagonally downward. The calculation of the median q_TIR in a given bin of luminosity should correct for measurements that in truth belong to a fainter luminosity range. To account for this, we construct broad luminosity bins (Δlog(L)≈ 1) and assume that most of the censored measurements would come to lie in the next lower luminosity bin (anything fainter would imply that they are more than 3 σ outliers to the IR–radio relation). We can then "average" away the effect of falsely assigned measurements by (1) computing the median 〈q_TIR〉 in two sets of luminosity bins which are offset by half a bin width and then (2) averaging the two estimates of the median thus obtained and reporting the new value half way between the centers of the two involved bins along the luminosity axis. The medians themselves are calculated by applying survival analysis to the jointly IR- and radio-selected data as previously done in Sections 6.1–6.3.

The results of this procedure are shown in the larger two windows in Figure 22. Using the COSMOS data, we see no evidence of an increase in the IR/radio ratio at L_TIR ∼ 10¹¹ L_☉ as suggested by Rieke et al. (2009). We do detect a higher value of 〈q_TIR〉 in the brightest IR luminosity bin but this increase happens around L_TIR ∼ 10¹³ L_☉, similar to the results of Younger et al. (2009). It should be mentioned, however, that the methodology used by Rieke et al. (2009) to derive q_24, 0 differs significantly from the one used here in that it involves, for example, luminosity-dependent (and template-based) conversions of IRAS 25 μm flux densities to 24 μm MIPS equivalent values.

While no universal trend for variations of q with IR luminosity are detected in our sample we do find that q_TIR is a decreasing function of radio luminosity (see the lower-most window in Figure 22). The trend is consistent and increases rapidly at L_1.4 GHz ∼ 10²⁴ W Hz⁻¹. This could potentially be the effect of contaminating AGNs at high radio luminosities in our optically selected sample of SFGs. However, the fact that I. Morić et al. (2010, in preparation) see a similar trend in local SF, composite and AGN-bearing systems which have been classified based on the standard optical line emission ratios (Kauffmann et al. 2003; Kewley et al. 2006) suggests that the trend is genuine.

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The two narrower windows in Figure 22 show the variations of the dispersion of q_TIR with IR (top) and radio luminosity (bottom). In the low-redshift samples of Yun et al. (2001) and Bressan et al. (2002), an increase in scatter with IR luminosity is detected. In the present data a similar, albeit very weak, tendency is seen; the reduced accuracy of the L_TIR measurements of the high-z galaxies likely masks most of the trend if present. The plot of $\sigma _{q_{\rm TIR}}$ versus L_1.4 GHz, on the other hand, shows a clear increase in the scatter which starts to manifest itself at the same radio luminosity at which the strong decline of 〈q_TIR〉 sets in.

While in the previous sections we usually tacitly plotted data points from high-z sources, the fitting of evolutionary trends in Sections 6.1 and 6.2 was restricted to galaxies at z< 1.4. This corresponds to the redshift at which the 4000 Å break leaves the reddest Subaru band with deep coverage (Taniguchi et al. 2007), the z band. After z∼ 1.4 the break is constrained by the NIR data of the J, H, and K_s bands (McCracken et al. 2010; P. Capak et al. 2010, in preparation). These exposures of the COSMOS field, however, are 2 mag shallower and have gaps between filters, leading to large uncertainties in the photometric redshift estimates. Beginning from about z∼ 2.5 the Lyα (1215 Å) break enters the wavelength range covered by the ground-based photometry (Capak et al. 2007; Taniguchi et al. 2007). As a consequence, the accuracy of the photometric redshift improves to again σ(Δz/(1 + z))≃ 0.03.

In an assessment of ongoing spectroscopic follow-up observations of high-z sources in the COSMOS field, P. Capak et al. (2010, in preparation) find that photometric redshift estimates of genuine high-z sources may be scattered to low redshift due to confusion between the Lyα and 4000 Å break. Most of the confusion is due to regions of the Lyα forest which are not as opaque as expected and/or light from nearby foreground galaxies contaminating the apertures. Conversely, there is little evidence for any upward scattering of galaxies at low and intermediate redshift to z≳ 2.5. This implies that sources with photometric redshift estimates >2.5 represent, with high likelihood, a clean—albeit not complete—sample of high-z objects.

Our sample contains more than ∼140 sources at redshift z> 2.5, of which approx. 60% have direct detections at 1.4 GHz and in at least one MIPS filter. As far as we are aware, this is the largest sample of high-z sources so far, for which it is possible to study the IR–radio relation based on direct detections rather than flux limits. We must point out, however, that only 2% of the high-z sources have a direct detection at 24 and 70 μm while the SEDs of the remaining 98% are only constrained by a direct detection at 24 μm and an upper flux limit at 70 μm. Accordingly, the calculated values of L_TIR luminosities must be regarded as fairly rough estimates of the true IR luminosity of these sources as they are primarily based on measurements made at a rest-frame wavelength of ∼6 μm. Murphy et al. (2009a) caution that the IR luminosities of high-luminosity and high-redshift sources (L_TIR/L_☉>10^12.5; z> 1.4) are generally overestimated by a factor of ∼4 even after subtraction of a flux contribution from AGN. However, in view of the COSMOS study of Kartaltepe et al. (2010)—who, in the same range of IR luminosities, do not see this trend and instead report that IR luminosities based solely on 24 μm data tend to be underestimated in general—we refrained from applying any corrections to our data.

Bearing in mind these uncertainties we plot the TIR/radio ratios of our high-z sources in Figure 23 (left panel). For illustrative purposes the measurements of q_TIR are colored according to their probability Pr (SF). We caution, however, that this classification is based on the fiducial (u−K) cut used throughout the paper so far and that the evidence presented in Figure 9 indicates that this threshold is no longer appropriate at z≳ 3. In view of this we do not distinguish between SF systems and AGN for the high-z sources but use this global sample to derive the average IR/radio flux ratio. The right-hand side shows the distribution function of q_TIR which is broad (σ = 0.59±0.05) and has a median of 2.71^+0.09_−0.14. This value is in good agreement with the local measurement of Bell (2003, dashed line) and is almost identical to the average value of 2.76^+0.02_−0.08 we find for the COSMOS data in our lowest redshift bin in Section 6.2. The average IR/radio properties of our high-redshift sample—the most distant sources of which are detected when the universe was only ∼1.5 Gyr old—are thus very similar to those observed in the local universe. It is important to remember, however, that at z⩾ 2.5 the COSMOS data contains mostly extremely IR-luminous HyLIRGS (L_TIR>10¹³ L_☉) which are a very different kind of object than those encountered at z< 0.5 where the majority of our sources have 10¹¹ < L_TIR/L_☉ < 10¹² (see Figure 1).

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At z< 1.4 where both photometric and spectroscopic redshifts have a high accuracy we found no compelling evidence of an evolving IR–radio relation. The mean IR/radio flux ratios in this redshift range are $\overline{q}_{24,\,0}=1.26\pm 0.13$, $\overline{q}_{70,\,0}=2.23\pm 0.13$, and $\overline{q}_{\rm TIR}=2.57\pm 0.13$, where the first two are computed using K-corrected IR and radio flux densities. With the exception of the highest redshift bin, the median IR/radio ratios in the different redshift bins covering the range z∈ [0, 1.4] are offset from the averages $\overline{q}$ by at most half the dispersion in the local IR–radio relation. An alternative to testing the constancy of rest-frame IR/radio flux ratios derived using IR SEDs of local SFGs, is to form IR/radio ratios with observed flux densities (see Section 6.1.1). Variations with redshift of q_24, obs and q_70, obs in the SF population can then be compared with the changes that would be expected for redshifted local galaxies (e.g., Ibar et al. 2008; Seymour et al. 2009) in order to assess if the IR/radio properties of the latter are compatible with those of high-z systems. Our analysis has shown that the decline with redshift of observed and predicted IR/radio ratios are indeed in good agreement.

Since measurements of q₂₄ and q₇₀ have been carried out with flux-limited data sets which likely are subject to some selection band-related bias (see Sections 5.1 and 7.2) only our value of q_TIR lends itself to a straightforward comparison with local measurements. Bell (2003) find q_TIR = 2.64  ±   0.02 for a sample of local SFGs and show that this figure is in excellent agreement with the IRAS-based FIR measurement q_FIR = 2.34  ± 0.01 of Yun et al. (2001). The COSMOS measurements of the mean IR/radio properties of high-redshift galaxies thus are fully consistent with the local average. Furthermore, a subsample of highly redshifted (z> 2.5) COSMOS galaxies has a median 2.71^+0.09_−0.14 which also agrees well with both the COSMOS data at low redshift and independent local measurements.

Due to Malmquist bias, we detect systems with very differing star formation rates (SFRs) in the low- and high-redshift universe where only extreme starbursts of the HyLIRG class (SFR ≳10³M_☉ yr⁻¹) are visible. That starburst galaxies which we observe when the universe was just 10% of its current age follow the same IR–radio correlation as local galaxies runs counter to expectation which would predict higher-than-average IR/radio ratios for such systems. Lisenfeld et al. (1996) have shown that a strong (and prompt) enhancement of the magnetic field strength is required lest a deficit of synchrotron emission develop due to high inverse Compton losses of the cosmic ray electrons in the strong radiation field generated by the starburst.

On the other hand, it could be that precisely this inability to detect sources with SFRs of "normal" (disk) galaxies at higher redshift is hiding changes in the IR/radio ratios of the SF population. In particular, such changes are expected as soon as the energy density of the CMB exceeds that of the galactic magnetic fields, whereupon inverse Compton losses off CMB photons begin to dominate synchrotron emission. Based on the typical magnetic field strengths in spiral arms (a few μG), Carilli et al. (2008) estimate that this could be the case for regular disks as early as z∼ 0.5. The VLA-COSMOS and S-COSMOS observations are not sensitive enough to detect such systems on an individual basis at high redshift. It is possible, however, that the stacking analysis of Carilli et al. (2008) has detected the suppressed radio emission due to inverse Compton cooling in Lyman break galaxies which are on average about 40 times fainter than the sources of a similar redshift presented in this paper.

Regardless of the remaining uncertainties surrounding the maintenance of the local IR–radio relation out to high redshift the observational fact per se is an important confirmation of the central assumption used in studies that have computed the SF history of the universe (Haarsma et al. 2000; Seymour et al. 2008; Dunne et al. 2009; Smolčić et al. 2009) using deep radio surveys, namely that of an unchanging proportionality between SFR and radio luminosity. The constancy of the IR–radio relation not only implies that IR and radio measurements are equally good tracers of star formation out to high redshift. It also suggests that the physical processes of massive star formation when the universe was only 1–2 Gyr old used to be strikingly similar to those at play in the local universe.

Currently available observations are not yet capable of revealing the exact workings of the astrophysical mechanisms that cause galaxies at intermediate and high redshift to lie on the IR–radio relation. The improvements in sensitivity and spatial resolution with Herschel and the extended Very Large Array (EVLA) will be a crucial step forward in this respect. A better sampling of the FIR emission from distant (U)LIRGs will reveal if their SEDs indeed match those of their local analogues as this study and previous work generally assume. Similarly, a systematic study of the range of radio spectral indices in SFGs is important to quantify the inaccuracies that are introduced by assuming a single generic value. Clarifying both these issues is essential if, e.g., trends for a luminosity dependence of $\overline{q}$ (see Rieke et al. 2009, and also our Figure 22) are to be put on a solid basis.

Apart from SFGs, the IR–radio relation is also observed by many AGN-bearing and/or composite systems (e.g., Sanders et al. 1989; Marx et al. 1994; Sopp & Alexander 1991; Roy et al. 1998; Murphy et al. 2009a; Seymour et al. 2009), albeit with a larger dispersion (Condon et al. 1982; Obrić et al. 2006; Mauch & Sadler 2007). In the COSMOS sample, we also observe that AGNs and SFGs often have very similar values of q₂₄ and q₇₀. This is true not only for optically selected AGNs but also for X-ray detected sources, implying that the phenomenon is not merely an artifact of our statistical color criterion for the separation between AGNs and SFGs. It should be emphasized that the finding is not an artificial consequence of template fitting; the abundance fraction of AGNs and SFGs has been derived using observed IR/radio ratios and hence does not involve any assumptions about the value of the radio spectral index or the shape of the IR SED.

We find that our optically selected AGNs and SFGs populate the locus of the correlation in nearly equal proportions out to at least z∼ 1. It is important to bear this in mind when IR/radio ratios are used—possibly in combination with other indicators—to distinguish between AGN and SF sources (e.g., Donley et al. 2005; Park et al. 2008; Seymour et al. 2008): radio-excess outliers can only be used to single out radio-loud AGN rather than a complete AGN sample and, conversely, selecting only objects that follow the correlation will result in a mixture of radio-quiet AGNs and SFGs rather than a pure SF sample. The fact that an AGN is present does not exclude coeval star formation in the host galaxy (e.g., Silverman et al. 2009). However, whether or not it is responsible for the similarity in IR/radio properties of SFGs and AGNs is still debated, with both supporting and contesting evidence being advanced (e.g., in the case of PG QSOs: see Sanders et al. 1989; Barthel 2006).

Even though the current data set has been selected both in the IR and radio, the fact that the average IR/radio ratios of the jointly selected sample are usually close to those of the IR-selected sample suggests that the average values 〈q〉 we quote are not the "intrinsic" value one would hope to find in an entirely unbiased sample. By separately studying an IR- and radio-selected sample of SFGs, however, it is at least possible to bracket the unbiased average IR/radio properties. As expected the jointly selected sample always lies within this region (with the exception of one case—see Table 6—where the amount by which the median of the jointly selected sample exceeds that of the IR-selected sample is still smaller than the width of the bins used for the calculation of the distribution functions of q in the doubly censored data set).

The shift between the average IR/radio properties of an IR- and a radio-selected sample are in principle predictable based on the dispersion of the IR–radio relation and the slope of the differential source counts (see Equation (5)). While comparable to the locally measured dispersion at intermediate redshift, the scatter of the relation in our COSMOS sample is significant at z> 1 where it is twice as large as at low redshift. Part of the increase could be due to an intrinsically higher scatter σ_q at high luminosities (Yun et al. 2001; Bressan et al. 2002) but probably is also a consequence of the limitations that are inherent in the calculation of IR luminosities when the IR SED is constrained by few points.²⁵ Under the simplified assumption of Euclidian source counts (β = 2.5), Equation (5) predicts the offset Δq_bias≈ 0.35 dex, found in our lowest redshift bins. It also makes a fair prediction of a shift of ∼0.7 dex between the IR- and radio-selected sample at z∼ 1 if one accounts for the larger scatter and the finding that at faint fluxes IR (Chary et al. 2004; Papovich et al. 2004) and radio (Richards 2000; Fomalont et al. 2006; Bondi et al. 2008) source counts are sub-Euclidean.²⁶

Figures 13, 15, and 17 show that apart from biasing the average IR/radio ratio, selection effects can also produce spurious evolution. Based on the radio-selected sample alone, we would infer a decrease of the mean 〈q〉 out to z∼ 1 (see also numerical values in Tables 3, 5, and 6). As the only recent study, Seymour et al. (2009) measure an average value of q₇₀ for their radio-selected sample that is reduced by 0.25 dex at z∼ 1 with respect to low redshift. This is close to the evolution d〈q₇₀〉/dz≈ 0.2-0.3 dex which we see for the radio-selected COSMOS sample. Based on the argument outlined above a simple explanation for this could be selection effects. However, as Seymour et al. (2009) adopt the different approach of stacking radio sources that are not detected at 70 μm rather than including them in the analysis in the form of flux limits as we have done, other explanations cannot be ruled out.

As stated in Section 5.1 there is ample evidence from the results summarized in Table 2 that the offset predicted by Equation (5) not only occurs in our data but also can reconcile most apparently discrepant measurements of mean IR/radio ratios in the literature. The one exception to this generally encouraging agreement are the highly inconsistent radio stacking results of Boyle et al. (2007), Beswick et al. (2008), and Garn et al. (2009) who have all studied the mean q₂₄ as a function of IR flux. Garn et al. (2009) in particular pointed out that the field-to-field variation of the mean IR/radio ratio can be considerable. The prospects are good that the issue will soon be resolved with the aid of significantly deeper EVLA observations at the ∼μJy level that will even directly detect the radio emission of the faintest 24 μm sources.

Following the completion of the VLA-COSMOS Large Project (Schinnerer et al. 2007) which had mapped the entire 2 deg² COSMOS field at 1.4 GHz with the VLA in antenna configurations A and C for 275 hr, the central 0.84 square degrees were re-imaged during an additional 62 hr in configuration A in the spring of 2006. The resulting VLA-COSMOS "Deep Project" mosaic has a resolution (FWHM of synthesized beam) of 25 × 25 and a pixel scale of 035 pixel⁻¹. The mean rms sensitivity is ≲0.01 mJy beam⁻¹ at the center of the field and more than 60% (80%) of the field has an rms level better than 0.02 (0.03) mJy (see Figure 3 in Schinnerer et al. 2010). The data reduction was carried out with AIPS (Greisen 2003) and followed standard procedures described in Schinnerer et al. (2007, 2010).

The AIPS source/component finding task SAD was used to detect radio sources in the Deep Project mosaic in multiple iterations with cutoff levels of successively lower S/N. The resulting source list was then combined with the VLA-Large Project catalog (Schinnerer et al. 2007) to create a list of 1.4 GHz sources (henceforth referred to as the "Joint catalog") that are at least 5 σ above the local background. Of the ∼2900 sources in the Joint catalog 51% are found to be unresolved. In these cases, the integrated flux density is set equal to the peak flux density of the sources which was measured with the AIPS task MAXFIT. In order to correct for bandwidth smearing, a position-dependent correction based on a model sensitivity map of the Deep Project mosaic has been applied to the peak flux values. Integrated flux densities (which are not affected by bandwidth smearing; e.g., Bondi et al. 2008) for resolved radio sources were determined by integrating over the size of the best-fitting elliptical Gaussian component with the task JMFIT. For another 131 (4.6%) of the sources in the catalog which were best fit by the sum of more than one Gaussian flux component, the task TVSTAT was used to measure the flux within a manually defined contour around the source. The distribution of 1.4 GHz flux density values in the Joint catalog is shown in Figure 15 of Schinnerer et al. (2010) where a detailed description of the construction of the catalog is given. The flux measurements carried out on the VLA-COSMOS radio maps have been compared to those of about 300 sources also observed at 1.4 GHz in the context of the NVSS and FIRST surveys (Condon et al. 1998; White et al. 1997). The agreement between the different data sets is reasonable except for a number of NVSS sources where the VLA-COSMOS observations have probably resolved out a large extended flux component (see Schinnerer et al. 2007).

For the multiple component sources which often have a complicated radio morphology due to outflows or continuum emission from star formation activity we set the most likely source position by visual inspection of HST ACS images (Koekemoer et al. 2007). In all other cases the source position is located at the peak of the radio emission as determined by MAXFIT. This definition was also adopted for a small fraction of multiple component sources that could not be reliably associated with an optical source. The typical accuracy of the radio positions is 013 as shown by Schinnerer et al. (2007).

While the Joint catalog serves as the basis of our radio-selected sample (introduced in Section 2.3.1) we will also use the MIPS 24 μm catalog described in Appendix A.2.1 to construct a sample of IR-selected sources. In addition to counterparts with S/N⩾ 5 from the Joint catalog, we also allow for sources having S/N> 3 in the presence of IR detections. To find additional sources in this S/N range, we ran MAXFIT at the position of IR detections without a counterpart in the Joint catalog. The MAXFIT box size is chosen in accordance with the uncertainty in the position of the IR source (see Appendix B). Since the vast majority of sources at these low detection levels are likely unresolved (radio sources with flux density ≲0.1 mJy typically have sub-arcsecond sizes, see Bondi et al. (2008) and references therein) their integrated flux is equated to the peak flux density.

If the extracted peak flux density has a significance level of less than 3 σ or if MAXFIT fails to converge, upper limits on the total flux from the local sky background are used as flux constraints for a potential source at that position. The noise in the Deep Project image was estimated with the AIPS task RMSD in a box of dimensions 105''×105'' that was moved across the survey area in steps of 245 in R.A. and decl. We adopt an upper flux limit of three times the local rms noise level for the rest of the analysis for undetected radio sources. The numeric value of this upper bound is obtained by reading the noise image at the pertinent position.

A.2.1. MIPS 24 μm Data

A.2.2. MIPS 70 μm Data

The COSMOS photometry catalog is an i-band selected catalog with PSF-matched photometry (FWHM = 06) from 30 broad, medium, and narrowband filters extracted at the positions of the Subaru i⁺-band detections. The wavelength range covered by these observations extends all the way from the UV at 1550 Å to the MIR at 8 μm. Capak et al. (2007, 2008) provide a complete description of the observations and data reduction leading to the compilation of the multiwavelength data set. Here we use the photometry catalog compiled by Ilbert et al. (2009) which lists more than 600,000 COSMOS galaxies with i⁺≲ 26 detected in a region roughly contiguous with the area covered by the VLA-COSMOS observations.

The COSMOS field has been observed with XMM-Newton for a total of ∼1.5 ×10⁶ s (∼400 hr), resulting in a homogeneous depth of ∼5 ×10⁴ s (Hasinger et al. 2007; Cappelluti et al. 2007, 2009) over 1.92 deg². The associated catalog includes 1887 point-like sources chosen to have a high probability of being a reliable detection²⁷ in at least one of the soft (0.5–2 keV), hard (2–10 keV), or ultra-hard (5–10 keV) bands down to limiting fluxes of ∼5 ×10⁻¹⁶, ∼3 ×10⁻¹⁵, and ∼5 ×10⁻¹⁵ erg cm⁻² s⁻¹, respectively (see Cappelluti et al. 2009 for additional details). Moreover, the central part of the field has been observed with Chandra (Elvis et al. 2009), providing precise positions for about half of the XMM-Newton sources. M. Brusa et al. (2010, in preparation) have determined optical counterparts to more than 90% of the XMM-Newton catalog sources. In the present work, we use only those with unique/secure optical counterparts and a successful fit to the UV to MIR SED (Salvato et al. 2009). The optical position has been used to associate our radio and IR sources to XMM-Newton detections using the same search radii as adopted for the band merging of our radio and IR sources with the optical photometry (see Appendix B).

The catalog of Ilbert et al. (2009) contains photometric redshift estimates for all tabulated objects with the exception of (1) sources lying in a "masked" region of the optical imaging due to a bright star in their vicinity or (2) sources that are detected at X-ray wavelengths, in which case a special set of AGN-templates was employed to derive a photometric redshift (Salvato et al. 2009).

Spectroscopic data have already been gathered for more than 20,000 sources in the COSMOS field (e.g., Lilly et al. 2007, 2009; Trump et al. 2007, 2009; J. Kartaltepe et al. 2010, in preparation; M. Salvato et al. 2010, in preparation). Whenever possible, we give precedence to redshift information from spectroscopy. In all other cases, we assign a photometric redshift to the sources in our radio- and IR-selected samples. The details of the band merging of the radio and IR catalogs with the optical data are presented in Appendix B, while the following two paragraphs summarize the properties and screening of the available redshift information itself.

A.5.1. Spectroscopic Redshifts

A.5.2. Photometric Redshifts

• 1.  
  IR-selected sample. The ∼6'' FWHM of the 24 μm PSF and Spitzer pointing uncertainties result in a non-negligible uncertainty on the source centroids of 24 μm sources. In order to have a more precise reference for the positional matching with other catalogs we searched for IRAC counterparts to each 24 μm source (the FWHM of the IRAC PSF is about 1/3 of that at 24 μm). Since the IRAC imaging of the COSMOS field is shallower than that performed at optical wavelengths but still sufficiently deep to detect a counterpart for most 24 μm sources this approach simultaneously reduces the likelihood of assigning the wrong optical counterpart in the subsequent band merging with the optical photometry catalog (see the last paragraph of this section).We correlated the 24 μm catalog with the IRAC catalog (Sanders et al. 2007; Ilbert et al. 2010) using a matching radius of 2'' and found 45,827 unambiguous counterparts (92% of all sources in the 24 μm catalog). In 572 cases more than one IRAC source was found within the search radius. Unless the closest of the potential counterparts was at least twice as close to the 24 μm position as the other candidates these sources (1% of the catalog) were excluded from the sample. In the case of 3668 24 μm sources for which no IRAC counterpart could be determined we searched the photometry catalog for optical counterparts within 06, the FWHM of the ground-based imaging. If this match was ambiguous the source was disregarded. Unique optical counterparts were found for another 1386 sources, however, and the final percentage of 24 μm sources with reference coordinates thus increased to 94%.
• 2.  
  Radio-selected sample. The 1.4 GHz source positions are accurate to within about 01, which is sufficiently accurate that they can be used directly as the reference position for the identification of optical and IR counterparts.

Starting with the radio source position we identified 2233 objects (77% of the radio catalog) that have a single 24 μm counterpart within 2'' while 647 (22%) radio sources remain unmatched. Based on their radio properties the 24 μm non-detections are primarily tbd. For <1% of the radio-selected sample the match was ambiguous. This subset includes two pairs of radio sources which have been assigned the same 24 μm counterpart and 17 for which more than one IR-detection was present within the search radius (mostly due to a complicated 24 μm morphology or the occasional over-deblended source).

Since the identification of 24 μm counterparts to 1.4 GHz sources in the immediately preceding paragraph was based on the precise radio positions these pairs of objects can be taken over directly into the IR-selected sample.³⁰ Note that subsequently we search for additional radio detections with S/N_1.4 GHz⩾ 3 in the vicinity of all 24 μm sources without a radio counterpart in the radio catalog described in Appendix A.1. The box in which we check for these additional faint radio sources has dimensions 175 × 175 (i.e., 5 × 5 pixel² at the pixel scale of the Deep Project image) which corresponds to the FWHM of the PSF of the first two IRAC bands. Radio fluxes for an additional 639 24 μm sources could be measured in this way.

We allow for a maximal separation of 6'' between reference coordinates and an accepted 70 μm source. Due to the broad FWHM (186) of the MIPS 70 μm PSF there is a relatively large probability that the emission from more than one 24 μm or 1.4 GHz source is blended into a single 70 μm source with no distinguishable secondary peaks by which the individual components could be separated. This occurred for ∼7% of the 1.4 GHz sources with a potential counterpart at 70 μm. In the IR-selected sample 21% of the matched sources had a potential 70 μm counterpart that lay within less than 6'' of at least one more 24 μm source. Unless these ambiguous 70 μm associations are at least twice as close to the nearest 24 μm or radio source as to the next best candidate they have been excluded for the analysis of the paper. Ultimately, this was the fate of 70% (50%) of the radio (24 μm) sources with blended 70 μm photometry. In the interest of achieving a high accuracy of the measured IR/radio properties we prefer this approach to, e.g., attempting to fit the 70 μm flux distribution with two components with peak positions fixed to the reference position.

To summarize, in we have found ∼1500 reliable 70 μm counterparts to 24 μm sources. A further 60 sources in the IR-selected catalog had more than one counterpart within 6'' and were removed together with the 24 μm objects that are associated with a blended 70 μm detection. The majority of the 24 μm sources (67%) has no directly detected counterpart. In the radio-selected sample we were able to identify ∼820 unambiguous 70 μm counterparts while excluding ∼60 sources because they either were associated with a blended 70 μm source or had an ambiguous match. For some 70% of the radio sources no 70 μm counterpart could be found.

A search radius of 06 is used for the cross-correlation of the radio-selected sample with the COSMOS photometry catalog and the spectroscopy catalog. This figure corresponds to the FWHM of the PSF-matched ground-based photometry. Whenever more than one optical counterpart is found within this distance the according radio source is removed from the catalog. This choice was made because any photometric redshift derived for the source would necessarily be unreliable as both sources contribute to the measured flux. The same applies to spectroscopic observations which have a slit width of ∼1'' and are thus treated analogously because sources within 05 could potentially lead to ambiguous spectral features.

To find optical counterparts and spectroscopic information for sources in the IR-selected catalog we use a search radius of 1'' which accounts for the somewhat larger uncertainty in the predominantly IRAC-based reference positions. Also, we ease the rejection criterion for ambiguous matches somewhat in that we accept optical or spectroscopic counterparts which are more than twice as close as the other candidate(s), provided the next best candidate lies beyond 06.

To derive the underlying distribution f(q) of IR/radio ratios q for a sample in which measurements of q can be limited both from above and below³¹ due to non-detections either in the IR or radio band we use the method of Schmitt et al. (1993). Their approach is based on maximum likelihood estimation and requires no assumptions about the form of the true distribution of q (it is thus said to be non-parametric). If represented in the form of a cumulative distribution, F(q), the estimator is equivalent to the probability of measuring a value of q smaller than a given ratio (which we will denote by a capital Q):

$$\begin{equation*} F(q) \equiv P(q \le {\rm Q}). \end{equation*}$$

In the following we summarize the steps involved in the construction of F(q).

To begin with, the n data points $\mathcal {Q}_i$ are arranged in M bins³² with ascending bin centers Q_j>Q_j+1, j = 1, ..., M. Let d_j be the number of detections in the jth bin which spans the range [Q_j − dQ/2, Q_j + dQ/2[. For the jth bin we also define u_j, the number of upper limits smaller than Q_j and, in analogy, l_j as the number of lower limits exceeding Q_j. Note that in view of the definition of u_j and l_j it is useful to choose bin boundaries offset by −dQ/2 and +dQ/2 for the upper and lower limits, respectively (see Avni et al. 1980). This eases the computation of the vectors d, u, and l which can be easily constructed as slightly shifted histograms of the detections as well as of the upper and lower limits.³³

The value f_j of the distribution function f(q) in the jth bin is constrained by the probability

$$\begin{equation} P\sim \prod _{j=1}^{M} f_j^{d_j} \left(\sum _{k=1}^{j-1} f_k\right)^{u_j} \left(\sum _{k=j+1}^{M} f_k\right)^{l_j} \end{equation} \tag{ C1 }$$

of obtaining the triplet (d_j, u_j, l_j). Intuitively, Equation (C1) can be understood as the product of the a priori probabilities of a measurement in the jth bin being a detection (first term), an upper limit which—if detectable—would in truth be located in a lower bin (second term) or an upper limit in actually located in a higher bin (third term). Schmitt (1985) and Campbell (1981) have shown that by introducing the likelihood and by using the constraint that ∑^M_j=1f_j = 1 the probability in Equation (C1) can be used to derive a system of M coupled fixed point equations for each of the f_j:

$$\begin{equation} f_j = \frac{d_j}{n} + \sum _{k=j+1}^M \frac{u_k}{n}\left(\frac{f_j}{\sum _{m=1}^{k-1} f_m}\right) + \sum _{k=1}^{j-1} \frac{l_k}{n}\left(\frac{f_j}{\sum _{m=k+1}^M f_m}\right). \end{equation} \tag{ C2 }$$

Here n = ∑^M_j=1(d_j + u_j + l_j) is the total number of measurements in the sample. Equations (C2) can be solved by iteration starting with f_j = d_j/n and u_j = l_j = 0 and the cumulative distribution constructed according to

$$\begin{equation} F_j = \sum _{k=1}^j f_k, \qquad j=1,\ldots, M \end{equation} \tag{ C3 }$$

once convergence has been achieved. This was typically the case after approximately 30–70 iterations.

Various approaches to calculate the uncertainty associated with the doubly censored cumulative distribution function F(q) have been proposed (see, e.g., Zhu & Sun 2007). Here we use the analytical formalism of Turnbull (1974) which is based on the Fisher information matrix J with elements³⁴

$$\begin{eqnarray} J_{jj} &= \frac{d_j}{(F_j-F_{j-1})^2} + \frac{d_{j+1}}{(F_{j+1}-F_j)^2} + \frac{l_j}{(1-F_j)^2}-\frac{u_j}{F_j^2}, \nonumber\\ j &= 1,\ldots, M-1 \nonumber \\ J_{j,j+1} &= J_{j+1,j} = -\frac{d_{j+1}}{(F_{j+1}-F_j)^2}, \nonumber \\ J_{MM} &= \frac{d_M}{(F_M-F_{M-1})^2} + \frac{l_M}{(1-F_M)^2}-\frac{u_M}{F_M^2},\nonumber \\ J_{ij} &= 0, \quad{\rm for}\quad |i-j| > 1. \end{eqnarray} \tag{ C4 }$$

J is a symmetric Jacobi matrix in which only the diagonal and off-diagonal elements are non-zero. In the limit of large samples its inverse, V, is an asymptotically unbiased estimate of the covariance matrix of the F_j (j = 1, ..., M) such that one has

$$\begin{equation} \sigma _{F_j}^2 = {\rm var}(F_j) = V_{ii}^2. \end{equation} \tag{ C5 }$$

Following Feigelson & Nelson (1985) and references therein it is then possible to define 2 σ confidence intervals of the form

$$\begin{equation} F_j^{\rm exp(\pm 1.96\,\sigma _{F_j})}, \end{equation} \tag{ C6 }$$

which always lie in the permissible range [0, 1].

By applying the algorithm of Equation (C2) to singly censored data sets, it was possible to check whether its output agrees with that obtained using the formalism of Appendix C.2. We found that the variance in Equation (C5) needed to be doubled in order to achieve consistency of the confidence intervals in Equation (C6) with those computed with the algorithm specifically designed to deal with singly censored data. For the sake of consistency the uncertainties on the doubly censored distribution functions stated throughout the paper have thus been computed using twice the nominal value of Equation (C5).

It can be useful to convey information on the shape of a (cumulative) distribution in form of percentiles. Given F_j and the associated confidence intervals, the pth percentile is defined as that q for which F(q) = p/100. Similarly, the points where the curves defined by the upper and lower confidence intervals—given by Equation (C6)—of F(q) are equal to p/100 are used as a measure of the uncertainty on the sought percentile.

A formalism which is identical to or simplified with respect to that of the previous section can be used when dealing with single censoring. If, for instance, we consider a radio-selected sample its distribution of q is censored from the left and can be represented as

$$\begin{equation} \begin{array}{c} \dsty \mathcal {Q}_i \,{=}\, \hbox{ max($q^-_i$, $q_i$)} \\ \ms \dsty \delta _i =\left\lbrace \begin{array}{@{}l@{\quad }l} {-}1 & \mbox{if }q_i<q^-_i\\ \ms \dsty 0 & \mbox{if }q_i \in [q^-_i, q^+_i] \end{array}\right. \end{array} \end{equation} \tag{ C7 }$$

in analogy to Equation (7). The treatment of singly censored data has been discussed in detail in Feigelson & Nelson (1985) and Schmitt (1985) with an emphasis on astronomical applications. The former paper also describes how to transform left censored into right censored data, the treatment of which has been documented more frequently in the literature for historical reasons (see references in the two aforementioned articles). Here we simply would like to point out that Equations (C2), which constrain the distribution function of doubly censored data, remain valid in the case of single censoring. However, the fact that either all of the u_i or l_i as defined in Appendix C.1 equal zero allows the coupled system of equations to be solved analytically in terms of the cumulative distribution function F_j through the Kaplan–Meier product limit estimator (Kaplan & Meier 1958). To do so we follow exactly the prescriptions of Feigelson & Nelson (1985).